TRIPLES, ALGEBRAS AND COHOMOLOGY

G X has a section s: XU G EU , then EU is isomorphic to

Here similarly, if p: E

the product XU — Y U in the underlying category A, with the following maps as inverse

isomorphisms:

(pU,σ)

G XU — Y U G EU

„

EU

G A takes

σ and „ being de¬ned as follows. It is easy to show a la Lemma 1 that U : B

`

G X into Y U -principal objects EU G XU . Using the given

Y -principal objects E

section s, the two maps

pU ·s

G

G EU

EU

EU

G Y U such that σ —¦ pU ·

are equal when followed by pU . σ arises as the unique map EU

G XU ,

s = EU . „ is de¬ned by the composition (x, y)„ = y —¦ xs, for any maps x: A

G Y U . Taking the identities we can also write „ = πY U —¦ πXU s, where the π™s are

y: A

the projections of the product. To prove that (pU, σ) and „ are inverses of each other, one

G X splits in B itself, and

may as well assume, which will simplify the writing, that E

ignore the underlying category. Then (p, σ)„ = σ —¦ ps = E. Also „ (p, σ) = X — Y , since

„ (p, σ)πX = πX is easy to see by compatibility of the group operation with the projection,

and „ (p, σ)πY = πY since both of these maps operate in the same way on „ ps: X — Y

G E (using the simply-transitive character of the operation on Y ):

„ (p, σ)πY —¦ „ ps = „ σ —¦ „ ps

= „ (σ —¦ ps)

= „,

πY —¦ „ ps = πY —¦(πY —¦ πX s)ps

= πY —¦ πX s

= „.

Of course, no assumption about the existence of products in the underlying category is

involved here. The lemma could be rephrased to assert that pU and σ have the universal

mapping property of projections.

A further remark: the reason for including the section in the structure of a principal

object and not merely assuming its existence, as is customary, is that this a¬ords us a

G 1-cocycles. Note that the postulated sections do not have

well-de¬ned map PO™s

to be preserved by maps of principal objects. Two principal objects identical except for

their sections will be isomorphic in the category PO(X, Y )·

Cohomology Classification of Principal Objects. We de¬ne a functor

˜

G Z 1 (X, Y )

PO(X, Y ) (relative to U )

where Z 1 (X, Y ) is the category of non-abelian 1-cocycles (homogeneous, for the moment)

described in our discussion of cohomology.

24 JONATHAN MOCK BECK

G X is a given principal

The functor ˜ is de¬ned in the following way. If p: E

object, consider the diagram

G

0

GE

s

XG2 G XG v

vvv

1

vvv

vvv p

7

X

s being the assumed section. Then ( 0 s)p = 0 (sp) = 0 = 1 = ( 1 s)p, so we know

G Y such that a —¦ 0 s = 1 s. A calculation which is

that there is a unique map a: XG2

given below shows that a is a non-abelian 1-cocycle. We de¬ne ˜ on objects by (E)˜ = a.

G E is a map in PO(X, Y ), form the diagram

If f : E

sf

G

GE

XGc cc s

cc

cc

cc p

cc

cc

1

X

G Y such that b —¦ s = sf . By

Since s p = = sp = sf p , there is a unique b: XG

G a . We de¬ne (f )˜ = b. One easily sees that

calculation b is a map of 1-cocycles a

˜ is a functor, the main veri¬cation needed being that (f f )˜ = b —¦ b , the product in the

G Y and the composition in the category Z 1 (X, Y ).

group of maps XG

Clearly ˜ induces a map

˜

G H 1 (X, Y )

PO(X, Y ) (relative to U )

where PO(X, Y ) is the set of connected components of the category PO(X, Y ). (Two

objects can be connected if there is a string of morphisms pointing in either direction

leading from one to the other. These morphisms become composable isomorphisms when

mapped into the groupoid Z 1 (X, Y ).)

Now suppose that the cartesian product X — Y exists in B. Because of the projection

G X and the left operation of Y on the second factor, X —Y is a Y -principal

πX : X —Y

G X — Y where 1: X G Y is

object over X. Its section of course is the map (X, 1): X

the neutral map. This principal object is trivial, or split. Any Y -principal object which is

split in B is isomorphic to X —Y . We refer to the component of PO(X, Y ) that X —Y lies

in as the trivial element of PO(X, Y ). (Not all principal objects in the trivial component

will be split, since PO(X, Y ) is not necessarily a groupoid.) Clearly

˜

G Z 1 (X, Y )

PO(X, Y ) (relative to U )

preserves the trivial object, which in Z 1 is the neutral 1-cocycle XG2 G Y . Thus the

induced map

˜

G H 1 (X, Y )

PO(X, Y )

25

TRIPLES, ALGEBRAS AND COHOMOLOGY

also preserves the trivial element, and another map is induced, denoted by

Aut(˜)

G H 0 (X, Y ).

Aut(X — Y )

The automorphism group is that of X — Y in the category of Y -principal objects.

Before proving that these maps ˜ are equivalences in the tripleable case, we give the

calculations necessary in order to know that the functor ˜ takes its values in the category

of 1-cocycles.

To prove that a = (E)˜ is a 1-cocycle, we let the maps 2 a —¦ 0 a and 1 a operate on

G E, and note that they give the same result.

the map 0 0 s: XG3

2a —¦ 0a —¦ 0 0s = 2 a —¦ 0 (a —¦ 0 s)

= 2a —¦ 0 1s

= 2a —¦ 2 0s

= 2 (a —¦ 0 s)

= 2 1 s,

1a —¦ 0 0s = 1a —¦ 1 0s

= 1 (a —¦ 0 s)

= 1 1 s.

G a is proved similarly. We let a —¦ 0 b and 1 b —¦ a operate

b = (f )˜ a 1-coboundary a

GE.

on 0 s : XG2

a —¦ 0 b —¦ 0 s = a —¦ 0 (b —¦ s )

= a —¦ 0 sf

= (a —¦ 0 s)f

= 1 sf,

1b —¦ a 0s = 1b —¦ 1s

—¦

= 1 (b —¦ s )

= 1 sf.

G AT G A, then ˜: PO

The tripleable case. When the adjoint pair is A

G Z 1 is an equivalence, implying the desired cohomology classi¬cation. The principal

objects considered are those trivialized by AT G A and the cohomology is also taken

GB G A is an

with respect to this underlying object functor. In general, if A

G Z 1 is compatible with the

arbitrary adjoint pair, one should still observe that ˜: PO

26 JONATHAN MOCK BECK

process of “tripleization,” in the sense of commutativity of the following diagram

˜

G Z 1 (X, Y )

PO(X, Y ) (relative to U )

Z 1 (¦)

PO(¦)

G Z 1 (X¦, Y ¦) (relative to U T )

PO(X¦, Y ¦)

˜T

Here ˜T is the functor which will be proved to be an equivalence, and ¦ is the “unit

of structure” (§1). (¦ is easily shown to map principal objects, after the manner of

GB G A can be

Lemma 3.) This diagram shows that an arbitrary adjoint pair A

G AT G A, wherein

canonically “closed up” via ¦, to a tripleable adjoint pair A

the interpretation of H 1 as isomorphism classes of principal objects always succeeds (as

GB G A is

well as the interpretation of H 0 as the hom functor). Of course if A

tripleable to start with (¦ an equivalence) then H 1 already classi¬es principal objects in

B.

G AT G A, X = (X, ξ) is a T-algebra,

Let us assume we are in the situation A

Y = (Y, θ) is a group object in the category of T-algebras, and let us write ˜ = ˜T .

THEOREM 5. If the cartesian product X — Y exists in A, then

˜

G Z 1 (X, Y ) (relative to U T )

PO(X, Y )

is an equivalence of categories, inducing isomorphisms

˜ G H 1 (X, Y )

PO(X, Y )

(relative to U T )

Aut(˜)

G H 0 (X, Y )

Aut(X — Y )

For algebra extensions the special case X = 1 (terminal object) is important (see

Theorem 6 below). In that case the product assumption can be dropped since 1 — Y Y.

The statement about Aut(X —Y ) above is related to Theorem 4. In general, Aut(X —Y )

(X, Y ) as groups since any principal object endomorphism of X — Y satis¬es (x, y)f =

G

(y —¦(x, 1))f = y —¦(x, 1)f and is therefore determined by (x, 1)f which is a map X

Y . Any endomorphism is thus an automorphism, although PO(X, Y ) as a whole is not

necessarily a groupoid.

G (X, ξ), with sections s: X

Proof. We consider (Y, θ)-principal T-algebras p: (E, ψ)

G E in A. What tripleableness does for us is allow us to express the structures of

principal algebras wholly in terms of the underlying category. We have the following

lemma, which is proved like Lemma 2.

p

G (X, ξ) is a (Y, θ)-principal T-algebra ⇐’ p: E G X is a Y -

LEMMA 4. (E, ψ)

G E satis¬es

principal object in A and the T-structure ψ: ET

(y —¦ e)T · ψ = (yT · θ) —¦(eT · ψ)

27

TRIPLES, ALGEBRAS AND COHOMOLOGY

G E, where y: A G Y , e: A G E are any maps in A.

in the set of maps AT

G (E , ψ ) is a map of (Y, θ)-principal algebras over (X, ξ) ⇐’ f is

f : (E, ψ)

a T-algebra map, preserves the projections into X, and satis¬es

(y —¦ e)f = y —¦ ef

G Y , e: A G E in A.

for any y: A

G Y in AT for

For the next step in the proof let us write, momentarily, a: XG2

G E in AT for the section. Then

the homogeneous 1-cocycle a = (E, ψ)˜, and s: XG

G Y correspond to a under

a is determined by the relation a —¦ 0 s = 1 s. Let a: XT

GE

adjointness. Then a is a nonhomogeneous 1-cocycle. The given section s: X

corresponds to s under adjointness also. What relation between a and s corresponds to

the relation between a and s? Now, in proving Lemma 4, one uses adjointness to push

the Y -operation on E down to the underlying category. One employs the following type